A celebrated theorem of G. Higman [Proc. R. Soc. Lond., Ser. A 262, 455–475 (1961; Zbl 0104.02101)] asserts that a finitely generated group \(G\) is recursively presented if and only if it is a subgroup of a finitely presented group \(H\). This theorem detects a deep connection between the logic concept of recursiveness (or Turing’s computability) and properties of finitely presented groups.
The Higman embedding \(G \hookrightarrow H\) makes the finitely presented group \(H\) quite large, saturating it with free subgroups. Therefore there were no nontrivial analogies of Higman’s theorem in proper subvarieties of the variety of all groups.
In the paper under review, an analogue of Higman’s theorem for the Burnside variety \(\mathfrak{B}_{n}\) of groups of odd exponent \(n\) (\(n\) sufficiently large) is proved. The main result is Theorem 1.1: There is a constant \(C\) such that for every odd integer \(n \geq C\), the following is true. A finitely generated group \(G\) satisfying the identity \(x^{n}=1\) has a presentation \(G=\langle A \mid \mathcal{R} \rangle\) with a finite set of generators \(A\) and a recursively enumerable set \(\mathcal{R}\) of defining relations if and only if it is a subgroup of a group \(H\) finitely presented in the variety \(\mathfrak{B}_{n}\).
The embedding property of Theorem 1.1 holds for groups \(G\) with countable sets of generators as well. Furthermore, the group \(H\) can be chosen with two generators. More precisely, the author proves Corollary 1.2: There is a constant \(C\) such that for every odd integer \(n \geq C\), the following is true. Let a group \(G\) from the variety \(\mathfrak{B}_{n}\) be given by a countable set of generators \(x_{1}, x_{2}, \ldots\) and a recursively enumerable set of defining relations \(\mathcal{R}\) in these generators. Then \(G\) is a subgroup of a 2-generated group \(E\) finitely presented in \(\mathfrak{B}_{n}\).
As applications of Theorem 1.1 and Corollary 1.2 the author reports the following examples:
(1) The direct product of all finite groups of exponent dividing \(n\) is embeddable in a 2-generated group finitely presented in the variety \(\mathfrak{B}_{n}\).
(2) There are infinite 2-generated, simple groups of large odd exponent \(n\), where all maximal subgroups have order \(n\) (in particular, every proper subgroup has order \(n\) if \(n\) is prime.) Moreover, there exist such “Tarski monsters” with decidable word problem by Theorem 28.4 in the author’s book [Geometry of defining relations in groups. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.20019)]. By Corollary 1.2, they are embeddable in 2-generated groups finitely presented in the variety \(\mathfrak{B}_{n}\).
At least two other consequences of the main theorem must be reported here.
Corollary 1.3: For every sufficiently large odd integer \(n\), there exists a 2-generated finitely presented in \(\mathfrak{B}_{n}\) group \(E\) containing, as subgroups with pairwise trivial intersections, isomorphic copies of all recursively presented groups \(\big \{ G_{i} \big \}_{i=1}^{\infty}\) of exponent \(n\).
Corollary 1.4: For every sufficiently large odd integer \(n\), there exists a group which is 2-generated, finitely presented in \(\mathfrak{B}_{n}\) and which has undecidable word problem.
At the end of the paper the author has placed a subject index that provides a great help to the reader in understanding the technical terms he used.